Apple TV 4K will not play purchased music

Welcome to the Apple Support Communities!

I understand you are trying to play music on your Apple TV. If you are in the Music app on your Apple TV you should see all of your purchased music in the Library. Is it a specific song or album that is not playing, or do none of the music in your Purchased playlist play?
Apple TV User Guide - Music - Library

Library includes any Apple Music content and playlists you added, iTunes purchases and playlists you created, and music you make available through iTunes Match.

First, update your tvOS to the latest version. In Settings , go to System > Software Updates and select Check for Update.
Then restart your Apple TV by go to Settings > System > Restart.
If you do not have an Apple Music subscription, you can still listen to music purchased on your computer by turning on Home Sharing:
Use Home Sharing to share iTunes content with other devices​
Take care.